A Length Function for Weyl Groups of extended affine root systems of Type $A_1$

TL;DR

This paper investigates the length function in Weyl groups associated with extended affine root systems of type A1, introducing a root basis and deriving explicit length calculations based on a unique element expression.

Contribution

It introduces a root basis for these root systems and provides explicit formulas for the length function in the Weyl group.

Findings

Defined a root basis for extended affine root systems of type A1

Derived explicit length formulas for Weyl group elements

Established a unique expression for Weyl group elements

Abstract

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type $A_1$. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.